Electrical – How does feedback make a system stable even in the presence of an inherently unstable element in that

analogcontrolcontrol systemlaplace transformstability

Consider the transfer function of a block being $$\frac{1}{s-0.5}$$ which is unstable in open loop. If we go for a feedback, magically it becomes stable even with the presence of the inherently unstable block. My thoughts on these are

  1. When an element is unstable , it doesn't mean it is unstable for all inputs. For certain inputs the block can still produce stable outputs.
  2. During feedback, the input given to the the block gets modified such that output doesn't blow up.

My questions are

  1. Are the above observations correct?
  2. The impulse response of the block is $$ e^{0.5t}u(t) $$ What the block does is, take the area under the signal at present instant and blow it up for subsequent instants. If so, a positive valued input should blowup the output exponentially. Then why is the output not blown up (exponentially) in following case, shown in figure?

The signal shown as input is the actual error signal when we use the block in a closed loop format with unit step input.enter image description here

Best Answer

"When an element is unstable , it doesn't mean it is unstable for all inputs. For certain inputs the block can still produce stable outputs."

At first, an "element" (a part) cannot be unstable. It is better to use the term "active block" or "amplifier". Hence, an amplifier can be unstable if it has feedback, which produces a closed-loop pole (pair) with a positive real part. This is true for all inputs because it is the feedback loop which produces self-excitement of the system - not the input signal.

"During feedback, the input given to the the block gets modified such that output doesn't blow up."

An unstable system can be stabilized if the overall negative feedback overrides (dominates) the positive feedback which would cause instability (without negative feedback). In your case (example), the gain block has an inherent positive feedback (causing a pole at +0.5). This gain block ist stabilized using - as shown - 100% negative feedback causing a total closed-loop gain of "2 (and a pole at "-0.5")